Punctured local holomorphic de Rham cohomology

Let V be a complex analytic space and x be an isolated singular point of V. We define the q-th punctured local holomorphic de Rham cohomology H-h(q)(V,x) to be the direct limit of H-h(q)(U - {x}) where U runs over strongly pseudoconvex neighborhoods of x in V, and H-h(q)(U - {x}) is the holomorphic de Rahm. cohomology of the complex manifold U - {x}. We prove that punctured local holomorphic de Rham cohomology is an important local invariant which can be used to tell when the singularity (V, x) is quasi-homogeneous. We also define and compute various Poincare number (p) over tilde ((i))(x) and (p) over bar ((i))(x) of isolated hypersurface singularity (V,x).